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Properties

Label 2-2511-279.2-c0-0-1
Degree $2$
Conductor $2511$
Sign $0.921 + 0.388i$
Analytic cond. $1.25315$
Root an. cond. $1.11944$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.669 − 0.743i)4-s + (1.58 + 0.336i)7-s + (0.564 + 0.251i)13-s + (−0.104 − 0.994i)16-s + (−0.5 − 0.363i)19-s + (−0.5 + 0.866i)25-s + (1.30 − 0.951i)28-s + (0.913 + 0.406i)31-s − 1.61·37-s + (−1.47 + 0.658i)43-s + (1.47 + 0.658i)49-s + (0.564 − 0.251i)52-s + (−0.309 + 0.535i)61-s + (−0.809 − 0.587i)64-s + (−0.309 − 0.535i)67-s + ⋯
L(s)  = 1  + (0.669 − 0.743i)4-s + (1.58 + 0.336i)7-s + (0.564 + 0.251i)13-s + (−0.104 − 0.994i)16-s + (−0.5 − 0.363i)19-s + (−0.5 + 0.866i)25-s + (1.30 − 0.951i)28-s + (0.913 + 0.406i)31-s − 1.61·37-s + (−1.47 + 0.658i)43-s + (1.47 + 0.658i)49-s + (0.564 − 0.251i)52-s + (−0.309 + 0.535i)61-s + (−0.809 − 0.587i)64-s + (−0.309 − 0.535i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2511 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 + 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2511 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 + 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2511\)    =    \(3^{4} \cdot 31\)
Sign: $0.921 + 0.388i$
Analytic conductor: \(1.25315\)
Root analytic conductor: \(1.11944\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2511} (188, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2511,\ (\ :0),\ 0.921 + 0.388i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.688362731\)
\(L(\frac12)\) \(\approx\) \(1.688362731\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 + (-0.913 - 0.406i)T \)
good2 \( 1 + (-0.669 + 0.743i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
7 \( 1 + (-1.58 - 0.336i)T + (0.913 + 0.406i)T^{2} \)
11 \( 1 + (-0.913 - 0.406i)T^{2} \)
13 \( 1 + (-0.564 - 0.251i)T + (0.669 + 0.743i)T^{2} \)
17 \( 1 + (0.809 + 0.587i)T^{2} \)
19 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
23 \( 1 + (-0.913 + 0.406i)T^{2} \)
29 \( 1 + (-0.669 + 0.743i)T^{2} \)
37 \( 1 + 1.61T + T^{2} \)
41 \( 1 + (0.978 + 0.207i)T^{2} \)
43 \( 1 + (1.47 - 0.658i)T + (0.669 - 0.743i)T^{2} \)
47 \( 1 + (0.978 - 0.207i)T^{2} \)
53 \( 1 + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.669 - 0.743i)T^{2} \)
61 \( 1 + (0.309 - 0.535i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.309 + 0.535i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.413 - 0.459i)T + (-0.104 + 0.994i)T^{2} \)
83 \( 1 + (-0.669 + 0.743i)T^{2} \)
89 \( 1 + (0.809 - 0.587i)T^{2} \)
97 \( 1 + (0.604 + 0.128i)T + (0.913 + 0.406i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.914962962555385560751940908798, −8.344197863850850742950794775017, −7.50621674808766072786864456244, −6.72748078079103592150777294965, −5.91176416453236707290518372500, −5.12942336504474689701989132452, −4.54113833929252468745623530758, −3.21857642171256342359257905515, −1.99710550974018003807792769525, −1.42555757454275371365964256123, 1.50586122130734639644104685869, 2.32852119252783965582319018029, 3.53943215074261444736504132553, 4.28592070860035690052709695946, 5.16349472016911064126751238200, 6.16751239597386800860967933571, 6.95023205799070096870702698125, 7.78065116061438275622941032839, 8.292857212608832636218967729158, 8.741938396652095943546583738100

Graph of the $Z$-function along the critical line